Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory

Carlos F. Lardizabal, Rafael R. Souza

Published 2015-10-12Version 1

Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Z\'u\~niga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence we obtain a quantum version of the classical probability result concerning the behavior of the columns (or rows) of the iterates of a stochastic matrix induced by a finite, irreducible, aperiodic Markov chain. We are also able to relate the ergodic property presented here with the notions of weak and uniform ergodicity known in the literature of noncommutative $L^1$-spaces. Together with a quantum trajectory approach we are able to examine a notion of hitting time and we see that many constructions, such as minimal solutions to hitting time problems, are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. As a more specific application we study the collection of walks induced by normal commuting contractions, for which the corresponding probability expressions are obtained. We examine open quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.